Rules for manipulating derivatives #121
Conversation
|
|
|
||
| --- | ||
|
|
||
| #### Inversion |
There was a problem hiding this comment.
| #### Inversion | |
| ``` {topic} Inversion Rule |
Putting the definitions of the rules for manipulating derivatives inside of {topic} directives will help them stand out. Do this for the chain rule as well.
| ```{math} | ||
| \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} |
There was a problem hiding this comment.
| ```{math} | |
| \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} | |
| \begin{equation} | |
| \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} | |
| \end{equation} |
Wrapping your equations inside \begin{equation}\end{equation} for single equations and \begin{align}\end{align} for multiple equations is preferred over the {math} directory. This will automatically number the equations which is useful for if you want to reference them later. Do this for all your equations.
| ``` | ||
|
|
||
| **Example:** | ||
|
|
There was a problem hiding this comment.
In this example, it is not entirely clear what the goal of these derivations is. Adding a short explanation at the start to about what the example is trying to find/prove/demonstrate will make it easier for the reader to follow along.
The goal according to the original notes is to calculate
The issue is that the direct derivation of
| \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} = \frac{2y}{z} | ||
| ``` | ||
|
|
||
| #### Chain Rule |
| ```{math} | ||
| \left( \frac{\partial x}{\partial y} \right)_z = \left( \frac{\partial x}{\partial w} \right)_z \left( \frac{\partial w}{\partial y} \right)_z | ||
| ``` |
There was a problem hiding this comment.
This equation could use some preamble to explain what the chain rule is and this new symbol
| \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} | ||
| ``` | ||
|
|
||
| **Example:** |
There was a problem hiding this comment.
Wrapping examples in the {example} directive will make the page look better organized. Do this for the other example as well.
| Then: | ||
| ```{math} | ||
| x = \frac{(w/z)^2}{z} = \frac{w^2}{z^3} | ||
| ``` |
There was a problem hiding this comment.
Here, you bring back the definition of x from the earlier example. This is fine, but you should try to be explicit when an example is using a quantity defined in previous examples. You could do this by including the previously used definition of x in a preamble of this example that goes something like "Given {{blank}} and {{blank}}, find {{blank}}." This will ensure the reader doesn't need to go back to the previous example.
Since using \begin{equation}\end{equation} should number all of your equations, you may also want to be explicit about which equations are being combined to make the math more clear. You might rewrite this line as "Then, using the definition of x from Equation {{put equation number here}}:"
| ```{math} | ||
| \left( \frac{\partial x}{\partial y} \right)_z = \frac{2w}{z^3} \cdot z = \frac{2w}{z^2} = \frac{2yz}{z^2} = \frac{2y}{z} | ||
| ``` |
There was a problem hiding this comment.
This should be broken up into multiple lines using \begin{align}\end{align}
New content
Resolves #30
Checklist
credits.md.